Journal of Power Sources 192 (2009) 486–493
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Journal of Power Sources
journal homepage: www.elsevier.com/locate/jpowsour
Global failure criteria for positive/electrolyte/negative structure of
planar solid oxide fuel cell
W.N. Liu a,∗ , X. Sun a , M.A. Khaleel a , J.M. Qu b
a
b
Pacific Northwest National Laboratory, P.O. Box 999, 906 Battelle Boulevard, Richland, WA 99354, United States
Georgia Institute of Technology, 801 Ferst Dr, Atlanta, GA 30332, United States
a r t i c l e
i n f o
Article history:
Received 29 January 2009
Received in revised form 3 March 2009
Accepted 3 March 2009
Available online 19 March 2009
Keywords:
SOFC PEN
Ceramics
Fracture mechanism
Global fracture criteria
Energy release rate
Critical curvature
a b s t r a c t
Due to mismatch of the coefficients of thermal expansion of various layers in the positive/electrolyte/negative (PEN) structures of solid oxide fuel cells (SOFC), thermal stresses and warpage
on the PEN are unavoidable due to the temperature changes from the stress-free sintering temperature
to room temperature during the PEN manufacturing process. In the meantime, additional mechanical
stresses will also be created by mechanical flattening during the stack assembly process. In order to
ensure the structural integrity of the cell and stack of SOFC, it is necessary to develop failure criteria for
SOFC PEN structures based on the initial flaws occurred during cell sintering and stack assembly. In this
paper, the global relationship between the critical energy release rate and critical curvature and maximum
displacement of the warped cells caused by the temperature changes as well as mechanical flattening
process is established so that possible failure of SOFC PEN structures may be predicted deterministically
by the measurement of the curvature and displacement of the warped cells.
© 2009 Elsevier B.V. All rights reserved.
1. Introduction
Fuel cells are high-efficiency energy conversion devices that are
environmentally friendly with little or no toxic emission. With high
efficiency and potential applications in stationary power generation
as well as auxiliary power units, solid oxide fuel cell (SOFC) continues to show great promise as a future power source. Among various
SOFC designs, anode-supported planar cells seem to provide the
best performance at the reasonable cost [1,2]. In anode-supported
cells, a porous anode provides the main structural support for the
thin (∼10 ␮m) yttria-stabilized zirconia (YSZ) electrolyte layer. The
cells must withstand a significant number of thermal cycles with
the build-up of internal stresses, which may lead to the cracking of
the electrolyte or delamination of the electrolyte–cathode/anode
interface [3].
Due to mismatch of the coefficient of thermal expansion (CTE)
of various layers in the positive/electrolyte/negative (PEN) structures of SOFC, thermal stresses on the PEN are unavoidable due to
the temperature changes from the stress-free sintering temperature to room temperature during the PEN manufacturing process.
The stresses in various layers, such as anode, electrolyte, and cathode, may cause the initiation and propagation of cracks. After the
sintering processes of PEN structure, cracks were observed in PEN
∗ Corresponding author. Tel.: +1 509 372 4967; fax: +1 509 372 4672.
E-mail address: wenning.liu@pnl.gov (W.N. Liu).
0378-7753/$ – see front matter © 2009 Elsevier B.V. All rights reserved.
doi:10.1016/j.jpowsour.2009.03.012
structures of SOFC. The cracks were generated between the electrolyte (YSZ) and fuel electrode (NiO–YSZ) in PEN structure of
fuel cells [4,5]. Such thermal internal stresses will also cause cell
warpage with a certain curvature. Although various techniques
have been developed for sintering the cells, some degree of cell
warpage remains after sintering. During stack assembly, the cells
with warpage will be flattened, which will contribute to the internal
stresses and the initiation and propagation of cracking and delamination. In [6], it was reported that processing-induced residual
stresses in the component layers of SOFCs can lead to fracture or
to cell curvature which impedes stack assembly. Generally, it has
been observed that there are two basic fracture mechanisms occurring in SOFC stacks: delamination along the interface and transverse
cracking across the layers [7]. Given the severe thermal cycles that
SOFCs must undergo, it is essential to develop suitable materials and
design stacks so that they are able to sustain thermo-mechanical
loads without cracking.
To ensure the structural integrity of the SOFC stack without
cracking and fracture, many efforts have been paid on investigating the fracture behavior of SOFCs and its PEN with the various
cracks and flaws by numerical analyses [8–10]. In the numerical
fracture analysis, pre-crack has to be formed in the various interfaces of PEN structures and inside various layers, and fine mesh
has to be used at the crack tips to capture the stress field of the
crack tips in order to obtain the accurate energy release rate. 3Dcurved crack in PEN structures was investigated by means of the
volume integrals model in [10]. It may be noted that the SOFC PEN is
W.N. Liu et al. / Journal of Power Sources 192 (2009) 486–493
487
Nomenclature
E
G
Gc
H
L
M
N
M
Q
Q
R
T
u
v
w
u0
v0
w0
W
Young’s modulus
energy release rate
interfacial fracture toughness
anode thickness
nominal cell size
flattening moment
resultant forces acting on all the layers
resultant moments acting on all the layers
elastic constant matrix
distributed transverse loading
curvature radius
temperature difference
layer displacement in the x-direction
layer displacement in the y-direction
layer displacement in the z-direction
displacement of the midplane in the x-direction
displacement of the midplane in the y-direction
displacement of the midplane in the z-direction
maximum allowable warpage
Greek symbols
˛
coefficient of thermal expansion
ˇ
Dundurs parameter
warped cell curvature
Poisson’s ratio
characterized by the multilayer thin film structure. A lot of efforts
have been contributed in the fracture mechanism of the thin film
structures. Interfacial fracture toughness of the thin film structures
was experimentally and theoretically studied in [11–16]. The channeling cracks of the thin film and the damage of its propagation
on the substrates were investigated deeply for elastic substrate and
elasto-plastic substrates in [17–21]. The detailed review is referred
to [22]. In these works, energy release rate and stress intensity factors are widely used as criteria of fracture failure, based on the
local stress state at the crack tip obtained by the theoretical and
numerical analyses.
While being an effective method for the reliability design
of SOFC PEN structures, the fracture toughness-based approach
described above will have difficulties in predicting failure induced
during the sintering and mechanical flattening processes for the
SOFC PEN materials. This is because the energy release rate may not
be measured directly for the PEN structures of SOFCs during sintering and flattening processes. To ensure the structural integrity and
reliability of the SOFC stack, it is necessary to develop a failure criterion for the initial failures that occur during cell–stack assembly.
In this paper, the global failure criteria for PEN fracture are established based on local fracture mechanics formulation for cracks at
various locations in the PEN that could be natural result on various interfaces and in cell materials due to the porous nature of
the anode and cathode. The critical energy release rate is related to
critical curvature and maximum deformation of the warped cells
during cell fabrication; therefore the allowable curvature of the
warped cells may be calculated, and possible structural failures of
the PEN structure can be predicted prior to stack assembly. Specifically, the global failure criteria are formulated based on the warpage
of the tri-layer PEN structure, i.e., the cell after sintering process.
The warpage of each cell is measured before stack assembly. Then,
using the global failure criteria, the engineers can predict whether a
cell can survive the stacking assembly process. Clearly, such global
failure criteria can also be used for designing the sintering processes to avoid excess warpage of the cells by controlling the pore
Fig. 1. A functionally graded multiple layer laminate.
size and porosity using the sintering temperature and time as well
organic compounds in the PEN structures. The purposes of developing such global failure criteria are to aid the initial design, materials
selection, and optimized assembly parameters of SOFC PENs.
2. Basic governing equations of thermal–mechanical
analysis of multi-layers and global fracture criterion
2.1. Basic governing equations of thermal–mechanical analysis
for functionally graded multi-layers
Consider a multilayered laminate of Lx × Ly , as shown in Fig. 1.
The cross-section of the laminate is schematically shown in Fig. 2.
The thermo-mechanical properties of each layer may vary layer by
layer.
Let u, v and w be the displacements in the x-direction, y-direction
and z-direction, respectively. Assuming classical plate bending theory holds and that plane cross-section remains plane, the strain in
the laminate can be written as [23]:
εx =
∂u0
∂u
∂2 w0
=
−z
,
∂x
∂x
∂x2
(1)
εy =
∂ v0
∂v
∂2 w0
=
−z
,
∂y
∂y
∂y2
(2)
xy =
∂u
∂v
∂u0
∂v0
∂2 w0
+
=
+
− 2z
,
∂y
∂x
∂y
∂x
∂x∂y
(3)
where u0 , v0 and w0 are the displacements of the middle plane z = 0.
The preceding strain–displacement relationship can be written in
terms of the midplane strains and the plate curvature as follows:
␧ = ␧0 − z␬
(4)
Fig. 2. Cross-section of a multilayer laminate.
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W.N. Liu et al. / Journal of Power Sources 192 (2009) 486–493
B=
n n N0 =
n M0 =
T
[ε0x , ε0y , ε0z ]
ε0 =
=
∂u0 ∂0 ∂u0
∂0
,
,
+
∂x ∂y ∂y
∂x
␬ = [x , y , xy ]T =
T
T
.
1
,
R
(5)
with radius R shown in Fig. 3, where W denotes the maximum
displacement of the multilayer laminate at the center.
Assuming each layer is linear elastic, isotropic, stresses in each
layer can be expressed as [24]:
␴ = Q(␧ − ˛T 1),
(6)
where
= [x , y , xy ]T ,
and 1 = [1, 1, 0]T .
and ˛ is the coefficient of thermal expansion (CTE) and T is the
temperature change. The above equation can be recast into the
following form:
␴ = Q(ε0 − z␬ − ˛T 1),
where
Q =
Q11
Q12
0
Q12
Q22
0
0
0
Q33
(7)
E
1 − 2
=
1
0
1
0
0
0
(1 − )/2
h/2
N=
␴dz,
␴zdz
∂Nxy
∂Nx
+
= 0,
∂x
∂y
=−
q + Nx
(8)
−h/2
∂2 w0
∂2 w0
∂2 w0
+ Ny
+ 2Nxy
2
2
∂x∂y
∂x
∂y
.
(12)
∂Mxy
∂Mx
+2
=0
∂x
∂y
(13)
Simply supported edge: For a simply supported edge, the deflection
and the moment are zero. Thus,
w0 (a, y) = 0,
Mx (a, y) = 0
(14)
Rigidly clamped edge: In this case, the deflection and the rotation
are zero. Thus,
w0 (a, y) = 0,
N = [Nx , Ny , Nxy ]T ,
M = [Mx , My , Mxy ]T .
N = Aε0 − B − N0 1,
(9)
M = Bε0 − D − M0 1,
(10)
where
n zi
zi+1
Q dz,
∂w0 (x, y) ∂x
= 0,
ε0x (a, y) = 0
(15)
x=a
2.2. Global fracture criteria
Substitution of (7) into (8) yields:
k=1
(11)
where q is the distributed transverse loading, and w0 represents the
z-displacement of the midplane.
With various boundary conditions, the close set of equations will
be obtained so that the curvature, displacement and stress in the
various layers may be calculated. The several boundary conditions
are listed as follows:
with
A=
∂Ny
∂Nxy
+
= 0,
∂y
∂x
∂4 w0
∂4 w0
∂4 w0
+2 2 2 +
4
∂x
∂x ∂y
∂y4
Mx (a, y) = 0,
h/2
M=
−h/2
zi
˛i TEi
z dz.
1−
Free edge: For a free edge, the bending moment and the shear force
must vanish. Thus,
with E and being the Young’s modulus and the Poisson’s ratio.
The resultant forces and moments acting on all the layers may
be calculated as
zi+1
˛i TEi
dz,
1−
Note that for functionally graded materials, both the elastic constant matrix Q and the CTE ˛ are functions of the coordinate z.
Furthermore, if the temperature distribution is not uniform across
the thickness, T is also a function of z.
Consider the equilibrium of an infinitesimal element in the laminate, the fact that the total forces and total moment acting on the
element must add up to zero provides the following equilibrium
equations [25]:
,
The curvature is defined as
=
zi+1
n k=1
∂2 w0 ∂2 w0 2∂2 w0
,
,
∂x∂y
∂x2
∂y2
Qz 2 dz,
zi
k=1
Fig. 3. Illustration of curvature radius R.
zi+1
zi
k=1
where
Qz dz,
zi
k=1
D=
zi+1
Assuming a crack/flaw A exists in a thin film structure as shown
in Fig. 4, thermal loading T and/or mechanical loading F will create
a corresponding stress field , where U represents the deformation
field, Ū is the given displacement at the boundary. G refers to the
energy release rate at the crack tips.
Stress increase caused by sintering and/or the stack assembly
processes may be calculated as
= f (F, E, X),
(16)
W.N. Liu et al. / Journal of Power Sources 192 (2009) 486–493
489
The plate curvature due to temperature differential T can be
obtained as [24]:
= ˝T,
where
˝=
N̄0 (B11 + B12 ) − M̄0 (A11 + A12 )
(B11 + B12 )2 − (D11 + D12 )(A11 + A12 )
with
N̄0 =
Fig. 4. Illustration of crack, load, stress and deformation.
k=1
where E and X represent the mechanical properties of materials and
geometry, respectively.
The energy release rate at the crack tip is denoted as G. It is a
function of both the mechanical properties E and the geometry X,
such that
G = g(F, E, X).
(17)
(18)
fracture failure occurs. It is a local fracture criterion based on the
stress field at the crack tip obtained either numerically, analytically,
or experimentally.
In the meantime, the thermal and/or mechanical load F induces
deformation of the structure, U, as
U = (F, E, X).
(19)
Combining Eqs. (18) and (19), the relationship between energy
release rate and deformation may be obtained as
U = (g −1 (G), E, X).
(20)
If the relationship given by Eq. (20) is monotonous, the critical
deformation Uc can be determined given the critical energy release
rate Gc ,
Uc = (g −1 (Gc ), E, X).
(21)
Therefore, fracture failure will occur if
U = (F, E, X) > Uc = (g −1 (Gc ), E, X).
M̄0 =
zi+1
zi
n k=1
zi
zi+1
(22)
Hence, the fracture criterion is based on the global variable, i.e., displacement field, which may be measured easily after the sintering
processes of the SOFC PEN structures.
(25)
˛i Ei
dz,
1−
˛i Ei
z dz.
1−
During stack assembly, the warped cells with curvature are
flattened by moment M. The relationship between the curvature and the flattening moment M is obtained as
= ˝M.
If this energy release rate exceeds its critical value, Gc ,
G = g(F, E, X) > Gc ,
n .
(26)
The corresponding stress increase caused by cell flattening can
now be calculated as
␴ = zQ␬
(27)
3.2. Global failure criteria for SOFC PEN
Due to the porous nature of anode and cathode in the PEN structures, initial flaws and crack initiation points will naturally exist on
the interfaces of anode/electrolyte/cathode and in the interior of the
materials. The types of initial flaws considered here are schematically illustrated in Fig. 5.
Although various techniques have been developed for sintering
the cell materials, certain degrees of cell warpage will remain after
sintering. During the stack assembly process, cells are flattened in
order to fit into the stack. Flattening exerts additional stresses to
the warped cells. Such stress may fracture the cell during the flattening process if the warpage is greater than some tolerable level.
Therefore, the global failure criteria developed here for PEN structure of SOFCs are based on local fracture mechanics formulations
for initial crack size at various location of the cell.
For most anode-supported, YSZ-based SOFCs, the crack types
A, B, C and D are particularly vulnerable to the flattening process.
The failure criteria below provide a first-order estimate of the maximum allowable warpage and maximum allowable curvature to
3. Global failure criteria for SOFC PEN
3.1. Thermal residual stress induced deformation of SOFC PEN
structures
During the sintering process, SOFC PEN structures are placed in
the furnace freely without any constraints on the boundary.
Assuming free edge condition for the PEN boundary at x = ±L/2
and y = ±L/2, the bending moment and the shear force must vanish,
which leads to the following set of boundary conditions [25] as
boundary condition given in (13):
L
Mx ± , y
2
= 0,
∂Mxy
∂Mx
+2
=0
∂x
∂y
Mx x, ±
L
2
=0
(23)
(24)
Fig. 5. Cracks at various locations in a cell, where A, surface crack in the cathode; B,
surface crack in the anode; C, interfacial crack between the cathode and electrolyte;
D, interfacial crack between the anode and the electrolyte; E, blister crack on the
anode/electrolyte interface; F, tunneling crack in the electrolyte
490
W.N. Liu et al. / Journal of Power Sources 192 (2009) 486–493
avoid fracture during the stack assembly process. In the following context, a is the crack length of each crack type, Ea(e,c) , a(e,c) ,
˛a(e,c) and ha(e,c) represent the Young’s modulus, Poisson’s ratio, CTE,
and thickness of the anode, electrolyte, and cathode, respectively.
In addition, subscript ce and ae stand for the combined properties
of cathode/electrolyte composite and anode/electrolyte composite,
respectively.
curvature is flattened by moment M, the relationship between and M can be obtained in Eq. (23) as
3.3. Crack A
=
First, we consider the critical curvature the tri-layer PEN structure can withstand during cooling from the sintering temperature,
i.e., stress-free temperature to room temperature given the initial
flaw A.
Based on elasticity theories, the average biaxial stress in the
cathode and electrolyte for temperature differential T can be
approximated as [22]:
where h represents the thickness of the PEN structure. Substitution
of Eq. (35) into (30) leads to
=
Ec
T˛,
1 − c
(28)
where
1 − c
,
Ec ˛
(29)
For stress in this crack configuration, the energy release rate may
be approximated by the following relations [21]:
G=
K2
,
Ec
(30)
where K, the stress intensity factor, is approximated for this type of
crack by
K = H
√
a
H = 1.1215 a 1 −
hc
1/2−s 1+
a
hc
.
Here a is the initial crack depth, is a fitting parameter which is
independent of a/h3 [21], and s is the root of the following equation
[26]:
cos(s) − 2
G=
h−a
Ec ,
2
h − a 2
2
ˇ1 − ˇ22
ˇ1 − ˇ2
(1 − s)2 +
= 0.
1 − ˇ2
1 − ˇ22
1 ˛
˝ 1 − c
(36)
h − a 2
+
2
Ec H 2 c2 .
(37)
2
˛
˝ (1 − c )
Ec H 2 c2 ,
(32)
+
h − a −1 1 G
c
2
H
Ec
.
(38)
3.4. Crack D
Considering the edge crack, Crack D, located in the interface
of the anode and electrolyte in the PEN structures. Again, crack
tip field generated by cooling is considered first. The cathode and
electrolyte in the PEN structure is considered as bi-layer composite plate. For bi-materials laminate, the equivalent elastic modulus,
Poisson’s ratio, and coefficient of thermal expansion, can be given
as [28]:
Ece =
he Ee + hc Ec
,
hce
(39)
ce =
he e + hc c
,
hce
(40)
˛ce =
˛e he Ee + ˛c hc Ec
.
he Ee + hc Ec
(41)
The energy release rate may be obtained under the temperature
differential as [16]:
G=
4hce
2
2
F1 (˛) (T ) ,
cce
F1 =
˝ 1 − c
c =
H ˛
Ec H 2 2 .
˝ 1 − c
where ˇ1 and ˇ2 are the two Dundurs parameters following the
work of Dundurs [27]. The relationship of the critical energy release
rate Gc and critical curvature c is obtained as
and
(35)
1 ˛
Gc =
with
Gc =
(34)
Superimpose the energy release rates from sintering and flattening, i.e., combining Eqs. (33) and (36), one obtains
(31)
1
.
Assuming free boundary conditions at the edge, the average stress
in the cathode may be obtained as
c =
˛a Ea ha + ˛e Ee he
.
˛a ha + ˛e he
Noted that T is the temperature difference from the stress-free
temperature to room temperature. Combining Eqs. (25) and (28)
yields
=˝
(B11 + B12 )2 − (D11 + D12 )(A11 + A12 )
For given critical energy release rate Gc , the critical curvature c ,
is then
˛ = ˛c − ˛ae ,
˛ae =
M(A11 + A12 )
=
C32 d12
I
+
d22
(42)
+ 12C3 d1 d2 d3 .
The parameters in the above equation are defined in Appendix
A as Ref. [26].
From Eqs. (25) and (42), one may obtain the relationship
between the energy release rate and the curvature as
2
G=
Gc
Ec
(33)
For given critical energy release rate Gc , the critical curvature
can now be determined.
Next, the energy release rate at the crack tip generated by the
flattening process during stack assembly is superimposed on that
resulted from the cooling process. Assuming a warped cell with
1 4hce F1 (˛) 2
,
cce
˝2
(43)
therefore,
=˝
1/2
cce
4hce F(˛)
2
√
G.
(44)
Next, crack tip field resulted by cell flattening during stack
assembly is derived. For a cell with curvature , the relationship
W.N. Liu et al. / Journal of Power Sources 192 (2009) 486–493
between the crack tip energy release rate and the curvature and is
expressed as
which leads to,
¯
=˝
16h3ce G
cce F2
Table 2
Values of the factor Y (×10−3 ).
Crack A
cce F2 2
,
¯2
16h3ce ˝
G=
491
(45)
Y
a = 0.01h3
a = 0.05h3
a = 0.1h3
a = 0.2h3
4.63
2.08
1.48
1.06
Crack C
Crack D
3.87
2.01
1/2
,
(46)
where
¯ =
˝
F2 =
A11 + A12
(B11 + B12 )2 − (D11 + D12 )(A11 + A12 )
C22
+
C32
I
,
+ 12C2 C3 d3 .
Again, combining the effects of cooling and cell flattening, the
relationship of the energy release rate and the curvature can be
obtained as
1
¯
˝
G=
1/2
cce F2
16h3ce
1
+
˝
4hce F1 (˛)
cce
2
1/2 2
2 .
(47)
Using similar derivation process, for each crack type illustrated
in Fig. 5, one may derive the relationship between the critical energy
release rate and critical curvature of the warped PEN structure as
c = Ȳ
Gc
,
he Ee
(48)
where Ȳ is dimensionless constant depending on the cell geometry,
material constants as well as the size and location of each crack type.
For Crack A,
Ȳ =
1 ˛
+
˝ 1 − c
h − a −1 1 h E
e e
2
H
Ec
,
and for Crack D,
Ȳ =
1
¯
˝
1/2
cce F2
16h3ce
1
+
˝
4hce F1 (˛)
cce
2
(49)
1/2 −1 he Ee .
(50)
Similarly, the relationship between the critical energy release
rate and maximum deformation of the warped PEN structure may
be obtained as
Wc
=Y
L
Gc
he Ee
L
he
.
Fig. 6. ␬c vs. Gc for the cell parameters given in Table 1.
anode. Therefore, crack types A, C and D may propagate during cell
flattening. The values of Y for these crack types are listed in Table 2.
For L = 10 cm, the relationship between the curvature and the energy
release rate is depicted in Fig. 6 for various crack types; similarly,
Fig. 7 illustrates the relationship between the maximum allowable
warpage W and the fracture toughness Gc for these cracks.
It is clear that the critical curvature and maximum allowable
deformation for the PEN structure with the surface crack on the
cathode will be lowered with the increment of the crack length
with given critical energy release rate. Compared with the interfacial edge crack between the anode and the electrolyte, the critical
curvature and maximum allowable deformation for the interfacial
edge crack between the cathode and electrolyte is much less. The
interfacial edge crack between the cathode and electrolyte and the
channeling crack in the cathode will be easier to result in fracture
failure of the PEN structure after the sintering process and during
the cell–stack assembly process. The crack and flaw size in the PEN
structure may be controlled by the sintering processes parameter
(51)
where W is the maximum allowable warpage, L is the nominal cell
size, Y, like Ȳ , is dimensionless constant, too, depending on the cell
geometry, material constants as well as the size and location of each
crack type.
4. Illustrative examples
As an example, consider a cell with geometry and material
parameters listed in Table 1. For this example, the normal stresses
are tensile in the cathode, and compressive in the electrolyte and
Table 1
Material properties and geometric parameters of a cell.
Cathode
Electrolyte
Anode
E (GPa)
CTE (×10−6 /◦ C)
Thickness (␮m)
90
200
96
0.3
0.3
0.3
11.7
10.8
11.2
75
15
500
Fig. 7. W vs. Gc for the cell parameters given in Table 1.
492
W.N. Liu et al. / Journal of Power Sources 192 (2009) 486–493
Department of Energy’s National Energy Technology Laboratory
(NETL). Pacific Northwest National Laboratory is operated by Battelle Memorial Institute for the U.S. Department of Energy under
Contract No. DE-AC05-76RL01830.
Appendix A
cce =
ce + 1
,
ce
ce = 3 − 4ce ,
1 = 3 − 41 ,
hce
,
h1
=
hce = h2 + h3 ,
Fig. 8. Relationships between critical curvature and displacement and critical
energy release rate.
and the component materials. The process parameter and component materials can be optimized with combination of the fracture
failure analysis developed here in order to gain the reliable PEN
structure.
For each crack existing in the PEN structure of SOFCs, for a given
interfacial strength (i.e., interfacial fracture toughness Gc ), the corresponding critical curvature c can be determined. This critical
curvature and maximum allowable deformation of the PEN structure will help to judge if PEN structure survives during the cell–stack
assembly process pursuing after the sintering process. For example,
as shown in Fig. 8 for Crack C, if the curvature and/or the maximum
deformation of the PEN structure induced by the sintering process
are larger than the critical value obtained here, fracture failure will
occur during cell–stack assembly (flattening). Therefore, curvature
and/or warpage of the PEN structure may be measured after the
sintering process, the PEN structure has to be abandoned if their
values of the curvature and/or warpage are larger than the critical
value obtained here.
5. Conclusions
In this paper, a suite of global failure criteria for the SOFC PEN
structure is derived based on fracture mechanics formulations of
local crack tip field for cracks at different locations with different
initial sizes. These failure criteria consider the stresses generated
from the initial PEN cooling from sintering as well as the mechanical
flattening process during stack assembly. The purposes of developing such global failure criteria are to aid the initial design, material
selection and optimization of SOFC fabrication process. The critical
energy release rate is related to critical curvature and maximum
deformation of the warped cells. Since the curvature and displacement of the warped cells can be measured prior to stack assembly,
engineers can use the global failure criteria developed here to predict possible failure of a cell structure for various initial crack types.
Such global failure criteria can also be used for designing the sintering processes such that allowable warpage and curvature level of
the PEN can be achieved for its survival in the subsequent assembly
process.
Acknowledgments
This paper was funded as part of the Solid-State Energy Conversion Alliance (SECA) Core Technology Program by the U.S.
ce (1 + 1) − 3 (ce + 1)
,
ce (1 + 1) + 3 (ce + 1)
=
=
1+
,
1−
ı=
1 + 2 + 2
,
2(1 + )
A0 =
1
+ ,
1
I0 =
3
1
3 ı−
C1 =
,
A0
C2 =
I0
C3 =
,
12I0
d1 =
1
1
−ı+ ,
2
1
−ı+
1
2
2
1
−3 ı−
ı
+1 +3
1
ı−
1
+ 3
,
,
d2 = 1 − C1 − C2 d1 ,
d3 = 2 (1 + ),
=
I=
1
,
1 + (4 + 62 + 33 )
1
.
12(1 + 3 )
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